ND Atomic Physics/04/06
Third-order many-body perturbation theory calculations
for the beryllium and magnesium isoelectronic sequences
H. C. Ho∗ and W. R. Johnson†
Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA
S. A. Blundell‡
Département de Recherche Fondamentale sur la Matière Condensée, CEA–Grenoble/DSM
17 rue des Martyrs, F–38054 Grenoble Cedex 9, France
M. S. Safronova§
Department of Physics, University of Delaware, Newark, Delaware 19716, USA
(Dated: June 23, 2006)
Relativistic third-order MBPT is applied to obtain energies of ions with two valence electrons
in the no virtual-pair approximation (NVPA). A total of 302 third-order Goldstone diagrams are
organized into 12 one-body and 23 two-body terms. Only third-order two-body terms and diagrams
are presented here, owing to the fact that the one-body terms are identical to the previously studied
third-order terms in monovalent ions. Dominant classes of diagrams are identified. The model
potential is a Dirac-Hartree-Fock V N −2 potential, and B-spline basis functions in a cavity of finite
radius are employed in the numerical calculations. The Breit interaction is taken into account
through second order of perturbation theory and the lowest-order Lamb shift is also evaluated.
Sample calculations are performed for berylliumlike ions with Z = 4–7, and for the magnesiumlike
ion P IV. The third-order energies are in excellent agreement with measurement with an accuracy
at 0.2% level for the cases considered. Comparisons are made with previous second-order MBPT
results and with other calculations. The third-order energy correction is shown to be significant,
improving second-order correlation energies by an order of magnitude.
PACS numbers: 31.15.Ar, 31.25.Md, 31.25.Jf, 31.30.Jv
I.
INTRODUCTION
The development of relativistic MBPT in recent
decades has been motivated in part by the need for accurate theoretical amplitudes of parity non-conserving
(PNC) transitions in heavy monovalent atoms such as
cesium and francium. Applications of the theoretical
methods developed to treat atomic PNC include support
of atomic clock development, tests of QED in precision
spectroscopy of highly-stripped ions, searches for time
variation in the fine-structure constant, and providing
precise astrophysical data.
Although nonrelativistic studies [1–7] and relativistic
HF calculations [8–10] for divalent atoms and ions have
been done for many years, only recently have relativistic many-body calculations been reported. As examples,
we note that all-order relativistic MBPT calculations for
transitions in berylliumlike ions with Z = 26 and 42 were
carried out by Lindroth and Hvarfner [11], while largescale configuration-interaction (CI) calculations for transitions in C III were performed by Chen et al. [12]. Rel-
∗ Electronic
address: [email protected]; present address: NCTS
(Physics Division), 101 Sec. 2 Kuang Fu Road, Hsinchu, Taiwan,
R.O.C.
† Electronic address: [email protected]
‡ Electronic address: [email protected]
§ Electronic address: [email protected]
ativistic many-body calculations for magnesiumlike ions
include the CI calculations of states in the n = 3 complex
by Chen and Cheng [13], and mixed CI-MBPT calculations of excitation energies in neutral Mg by Savukov and
Johnson [14].
Second-order relativistic MBPT was applied to Be-like
ions by Safronova et al. [15] and energies were found to
be accurate at the 2% level. In this paper, we extend relativistic MBPT for divalent atoms and ions to third order. We give a detailed treatment of the two-body terms
here; the one-body terms are identical to those for monovalent systems and are discussed in detail by Blundell
et al. [16]. The long-range goal of the present research is
to extend the relativistic singles-doubles coupled-cluster
(SDCC) formalism to atoms and ions with two valence
electrons. The present calculations permit us to identify
and evaluate those third-order terms missing from the
SDCC expansion.
II.
THEORETICAL METHOD
The model potential for our MBPT calculation is
the Dirac-Hartree-Fock V N −2 potential. The DiracCoulomb-Breit Hamiltonian for an N -electron atom is
H = H0 + V , where
H0 =
N
X
i=1
[hD (i) + u (ri )] ,
2
and the Dirac Hamiltonian is
hD = cα·p + βc2 + Vnuc (r),
where Vnuc is obtained assuming a Fermi nuclear charge
distribution. All equations are in atomic units. The perturbation is
¶ X
N µ
N
X
1
V =
u (ri ) ,
+ bij −
rij
i<j=1
i=1
where bij is the Breit interaction
"
#
1
(αi · rij ) (αj · rij )
bij = −
αi · αj +
.
2
2rij
rij
In the no virtual-pair approximation, the excitations are
limited to positive-energy eigenstates of H. [17–20]
The eigenstates of a divalent system having angular
momentum (J, M ) are described by the coupled states
X
ΦJM (vw) = ηvw
hjv jw , mv mw |JM i a†v a†w |0i , (1)
mv mw
where |0i represents the ground state of the ionic core
and ηvw is the normalization constant
½
1 for v 6= w
ηvw = √1 for v = w .
2
Here v and w specify the corresponding one-electron
states with quantum numbers (nv , lv , jv , mv ) and
(nw , lw , jw , mw ). The model space P is defined by the set
of total angular-momentum states (1); the model-space
projection operator is
X
P=
|ΦJM (vw)i hΦJM (vw)| .
JM
(v≤w)
The orthogonal-space operator Q is simply 1−P.
The wave operator Ω is found by solving the generalized Bloch equation [21]
[Ω, H0 ] P = (V Ω − ΩPV Ω)P.
The effective Hamiltonian is given in terms of the wave
operator
Heff = P H0 P + P V ΩP.
A.
Effective Hamiltonian
We first find the configuration-weight vector by di(0)
(1)
agonalizing the Hamiltonian H = Heff + Heff using total angular-momentum eigenstates (1) as basis.
Higher-order energies are obtained by operating the effective Hamiltonian of the corresponding order on the
configuration-weight vector. For simplicity, matrix elements of the effective Hamiltonian are given for the uncoupled states, |vwi ≡ a†v a†w |0i and |v 0 w0 i. In third order, each element consists of 12 one-body and 23 twobody terms. They represent a total of 84 one-body and
218 two-body Goldstone diagrams. The multiplications
of Clebsch-Gordan coefficients and the summations over
magnetic quantum numbers are carried out during angular decomposition. Only the two-body part of the third
order is discussed here as the one-body part and complete
second-order results are already presented in Refs. [16]
3
and [15].
T1 =
Z =
X
abcd
S1 =
S2 =
S3 =
S4 =
D1 =
D2 =
gcdwv gabcd g̃w0 v0 ab
(εab − εv0 w0 ) (εcd − εv0 w0 )
X
gacmw g̃mbac g̃v0 w0 vb
(εbv − εv0 w0 ) (εacv − εmv0 w0 )
abcm
˜´
`
ˆ
× 1 + v ↔ w, v 0 ↔ w0 + c.c.
X
g̃abwm g̃mcbv g̃v0 w0 ca
(εac − εv0 w0 ) (εabv − εmv0 w0 )
abcm
`
ˆ
˜´
× 1 + v ↔ w, v 0 ↔ w0 + c.c.
X
gabwm g̃w0 cab g̃v0 mvc
(εcv − εmv0 ) (εabv − εmv0 w0 )
abcm
`
ˆ
˜´
×(1 − [v ↔ w]) 1 − v 0 ↔ w0 + c.c.
X
g̃abwm g̃v0 cvb g̃w0 mac
(εac − εmw0 ) (εabv − εmv0 w0 )
abcm
`
ˆ
˜´
×(1 − [v ↔ w]) 1 − v 0 ↔ w0
X
gabmn g̃mnwb g̃v0 w0 va
−
(εbw − εmn ) (εav − εv0 w0 )
abmn
`
ˆ
˜´
× 1 + v ↔ w, v 0 ↔ w0 + c.c.
X 0 gabmn gmnvw g̃v0 w0 ab »
1
(εabvw − εmnv0 w0 ) (εvw − εmn )
–
1
+ c.c.
+
(εab − εv0 w0 )
X 0 g̃abmn gw0 nab g̃v0 mvw »
1
−
(εabvw − εmnv0 w0 ) (εab − εnw0 )
abmn
–
`
ˆ
˜´
1
+
1 + v ↔ w, v 0 ↔ w0 + c.c.
(εvw − εmv0 )
X
g̃abmn g̃w0 nwb g̃v0 mva
(1 − [v ↔ w]) + c.c.
(εbw − εnw0 ) (εav − εmv0 )
abmn
X
g̃w0 amn g̃nbaw g̃v0 mvb
−
(εavw − εmnv0 ) (εbv − εmv0 )
abmn
`
ˆ
˜´
×(1 − [v ↔ w]) 1 − v 0 ↔ w0 + c.c.
X
g̃w0 amn g̃nbvw g̃v0 mab
(εavw − εmnv0 ) (εab − εmv0 )
abmn
`
ˆ
˜´
× 1 + v ↔ w, v 0 ↔ w0 + c.c.
X
g̃abmw g̃w0 mbn g̃v0 nva
−
(εabv − εmv0 w0 ) (εav − εnv0 )
abmn
`
ˆ
˜´
×(1 − [v ↔ w]) 1 − v 0 ↔ w0 + c.c.
X
g̃w0 bwn g̃nabm g̃v0 mva
(εbv − εnv0 ) (εav − εmv0 )
abmn
`
ˆ
˜´
×(1 − [v ↔ w]) 1 − v 0 ↔ w0
X
gabwm g̃v0 mvn g̃w0 nab
−
(εabv − εmv0 w0 ) (εab − εnw0 )
abmn
`
ˆ
˜´
×(1 − [v ↔ w]) 1 − v 0 ↔ w0
abmn
D3 =
D4 =
D5 =
D6 =
D7 =
D8 =
D9 =
T2 =
T3 =
T4 =
Q =
B1 =
B2 =
B3 =
X
gw0 bmn g̃v0 avb g̃mnwa
(εbvw − εmnv0 ) (εaw − εmn )
abmn
`
ˆ
˜´
×(1 − [v ↔ w]) 1 − v 0 ↔ w0
X0
gw0 anm g̃mnar g̃v0 rvw
(ε
avw − εmnv 0 ) (εvw − εrv 0 )
amnr
`
ˆ
˜´
× 1 + v ↔ w, v 0 ↔ w0 + c.c.
X
gw0 amn g̃mnwr g̃v0 rva
(ε
avw − εmnv 0 ) (εav − εrv 0 )
amnr
`
ˆ
˜´
×(1 − [v ↔ w]) 1 − v 0 ↔ w0 + c.c.
X0
g̃w0 anr g̃v0 rma g̃mnvw
(εavw − εnrv0 ) (εvw − εmn )
amnr
`
ˆ
˜´
× 1 + v ↔ w, v 0 ↔ w0 + c.c.
X
g̃w0 arn g̃v0 nvm g̃rmwa
(ε
avw − εnrv 0 ) (εaw − εmr )
amnr
`
ˆ
˜´
×(1 − [v ↔ w]) 1 − v 0 ↔ w0
X 0 gv0 w0 rs grsmn g̃mnvw
(εvw − εrs ) (εvw − εmn )
mnrs
X
g̃w0 amn gnmax g̃v0 xvw
−
(ε
avw − εmnv 0 ) (εax − εmn )
amnx
`
ˆ
˜´
× 1 + v ↔ w, v 0 ↔ w0
X
g̃v0 axm g̃w0 mya g̃xyvw
−
(ε
avw − εmxw0 ) (εay − εmw0 )
amxy
`
ˆ
˜´
× 1 + v ↔ w, v 0 ↔ w0
X 0 gv0 w0 mn gmnxy g̃xyvw
,
−
(εvw − εmn ) (εxy − εmn )
mnxy
D10 = −
­ ¯ −1 ¯ ®
¯ kl is the Coulomb matrix element
where gijkl ≡ ij ¯r12
defined in terms of single-electron functions as
ZZ
1
φ† (r1 ) φ†j (r2 ) φk (r1 ) φl (r2 ) ,
gijkl =
dr1 dr2
|r1 − r2 | i
and g̃ijkl ≡ gijkl − gijlk . The notation εijkl ≡ εi + εj +
εk + εl , etc. for the sum of single-electron eigenenergies
has also been used. The third-order terms are arranged
by the number of excited states in the sums over states.
Zero, single, double, triple, quadruple excited-state terms
are designated by the letters Z, S, D, T, Q. Terms associated with backwards (folded) diagrams are designated
by B. Backwards diagrams are unique for open-shell
systems and exist only in the third or higher order of
MBPT. The summation indices (a, b, c, d) refer to core
states, (m, n, r, s) refer to excited states and, in backward diagrams, indices (x, y) refer to valence states. The
primes above the summation signs indicate that excited
states (m, n, r, s) are restricted to the orthogonal space
Q only. This restriction applies only to the term with
denominator 1/(εvw − εmn ) in D2 and to the term with
denominator 1/(εvw −εmv0 ) in D3 . The c.c. denotes complex conjugate. The conjugate diagrams are obtained by
a reflection through a horizontal axis, with the initial
and final states switched (vw) ↔ (v 0 w0 ). Direct diagrams of the two-body terms are shown in Fig. 1. Technically, there are subtle changes in energy denominators
4
in going from the term presented to its c.c. counterpart.
These changes can be deduced by re-drawing the diagram
upside-down and reading off the new denominators.
In D1 and D4 (but not in D2 and D3 ) we combined
diagrams associated with double excitations that have
the same numerators but different denominators using
the formula
1
1
1
+
=
.
(A + B)A (A + B)B
AB
Diagrams D1 – D4 are special in the sense that two orderings are possible. The ambiguous vertices are labeled by
crosses in the diagrams. Many of the third-order terms in
the two-body part have external exchanges and complex
conjugates so each diagram illustrated in Fig. 1 has from
one to eight variants. The largest fraction of computer
time is spent evaluating the term Q and most of the remainder is spent on repetitive evaluation of terms Dk , Tk
and their variants. Angular decompositions of the direct
formulas are listed in the Appendix.
III.
APPLICATION AND DISCUSSION
As a first illustration, we apply the theory described
above to obtain energies of the ground state and excited states in the n = 2 complex for Be-like ions. In
Table I, we give a detailed breakdown of the contributions from first-, second- and third-order perturbation
theory, together with corrections from the Breit interaction and from the Lamb-shift, for excitation energies of
(2s2p) 3P0,1,2 states of Be-like ions. The experimental energies are taken from the NIST database for atomic spectroscopy [22]. Energies E (0+1) represent the lowest-order
energies including the Breit correction. Lowest-order
Lamb shifts ELamb are obtained following the method described in Ref. [23]. We find that the residual differences
between calculated and measured energies ∆E decrease
rapidly with increasing Z. This is expected since MBPT
converges better for charged ions than for neutral atoms.
In fact, for highly-charged ions, correlations are expected
to decrease approximately as Z 2−n , where n is the order
of perturbation theory [24]. On the other hand, QED
effects (Lamb shifts) become more important along an
isoelectronic sequence. The results in Table I, confirm
both of these trends. Results of our calculations of excitation energies of all levels in the n = 2 complex for
Be-like N (N IV) are presented in Table II and are seen
to be in agreement with measurement to parts in 104 .
As a more involved example, we give a complete breakdown of contributions to energies of low-lying states in
the n = 3 complex for the Mg-like ion P IV in Table III.
For both N IV and P IV, correlations are seen to account
for about 10% of the total energies. For Be-like ions, the
third-order correlations are only 7–10% of the secondorder correlations. The second-order correlation energies
are an order of magnitude smaller than the corresponding DHF energies. The Breit correction B (2) , which is
TABLE I: Comparisons of third-order energies (cm−1 ) of the
triplet (2s2p) 3 P states of Be-like ions `Z=4–7
´ with measurements are given, illustrating the rapid 1/Z 3 decrease of the
residual correlation corrections with increasing Z. A breakdown of contributions to the energy from Coulomb and Breit
correlation corrections and the Lamb shift is given.
Z
4
E (0+1)
E (2)
B (2)
E (3)
ELamb
Etot
Eexpt
∆E
(0+1)
E
E (2)
B (2)
E (3)
ELamb
Etot
Eexpt
∆E
(0+1)
E
E (2)
B (2)
E (3)
ELamb
Etot
Eexpt
∆E
6
7
23607.9
-3114.3
-1.7
473.5
-0.9
20964.6
21978.3
-1014
5
(2s2p) 3 P0
39116.7
-2583.2
-3.7
598.0
-3.3
37124.4
37336.7
-212
54204.5
-2344.0
-6.7
412.9
-8.4
52258.4
52367.1
-109
69072.2
-2201.5
-10.5
294.9
-17.7
67137.3
67209.2
-72
23607.4
-3113.9
-0.6
473.4
-0.9
20965.5
21978.9
-1013
(2s2p) 3 P1
39120.2
-2582.2
-1.7
597.8
-3.2
37130.9
37342.4
-212
54223.4
-2342.2
-3.3
412.7
-8.3
52282.3
52390.8
-109
69127.7
-2198.6
-5.6
294.7
-17.5
67200.7
67272.3
-72
23608.2
-3113.0
0.4
473.2
-0.8
20968.0
21981.3
-1013
(2s2p) 3 P2
39132.6
-2580.1
0.5
597.4
-3.2
37147.3
37358.3
-211
54272.9
-2338.4
0.4
412.3
-8.2
52339.1
52447.1
-108
69260.8
-2192.6
0.0
294.1
-17.1
67345.2
67416.3
-71
obtained by linearizing the second-order matrix elements
in Breit interaction, is also small for such lightly-charged
ions.
Our calculations for Be-like ions are able to produce
results accurate to order of ten cm−1 . This shows that
the third-order energy correction is very important for
divalent ions. By comparison, the second-order results
of Safronova et al. [15] for N IV agree with experiment
at the level of a few hundred cm−1 . It should be noted
that results from the CI+MBPT method [14] mentioned
in the introduction are consistently more accurate than
the present third-order results. However, the CI+MBPT
calculations contain a free parameter in the energy denominators that is adjusted to give optimized energies.
In contrast, third-order MBPT is completely ab initio.
For the C III ion, the large-scale CI calculations of Chen
et al. [12] mentioned in the introduction also give tran-
5
TABLE II: Third-order energies (cm−1 ) of states in the n = 2 complex of the Be-like ion N IV, including corrections for the
Breit interaction and the Lamb shift.
3
Etot
Eexpt
∆E
P0o
67137.3
67209.2
-72
3
P1o
67200.7
67272.3
-72
3
P2o
67345.2
67416.3
-71
1
P1o
130764.1
130693.9
70
TABLE III: Comparison with measurement of theoretical energies (cm−1 ) of some of the low-lying states in the n = 3
complex of the Mg-like ion P IV, including a breakdown of
contributions from Coulomb and Breit correlation corrections
and the Lamb shift.
E (0+1)
E (2)
B (2)
E (3)
ELamb
Etot
Eexpt
∆E
(0+1)
E
E (2)
B (2)
E (3)
ELamb
Etot
Eexpt
∆E
(0+1)
E
E (2)
B (2)
E (3)
ELamb
Etot
Eexpt
∆E
(3s3p) 3 P0
67021.3
110.3
-0.9
807.4
-21.1
67917.1
67918.0
-0.9
(3s3p) 1 P1
120479.5
-20906.0
-15.8
6470.7
-20.7
106007.7
105190.4
817
` 2´ 3
3p
P1
166200.8
-2013.9
-4.8
1087.9
-43.8
165226.1
165185.4
41
(3s3p) 3 P1
67242.9
116.0
0.3
807.6
-20.9
68146.0
68146.5
-0.5
` 2´ 1
3p
D2
180554.7
-61699.8
-8.4
48769.6
-43.6
167572.4
166144.0
1428
` 2´ 3
3p
P2
166633.3
-2008.1
-2.7
1077.3
-43.4
165656.5
165654.0
3
(3s3p) 3 P2
67696.5
130.1
1.3
807.6
-20.5
68615.0
68615.2
-0.2
` 2´ 3
3p
P0
165971.6
-2027.7
-5.2
1089.6
-44.0
164984.3
164941.4
43
` 2´ 1
3p
S0
212201.4
-23060.7
-24.6
5810.7
-41.3
194885.6
194591.8
294
sition energies accurate to better than a hundred cm−1
on average. Those large-scale CI calculations are also ab
initio and have about the same accuracy as third-order
MBPT for states in C III.
For the P IV ion, our results are in good agreement
with experiment, the average discrepancy being several
hundred cm−1 . Chaudhuri et al. [25] employed an effective valence shell Hamiltonian (EVSH) to calculate energies of Mg-like ions and obtained results for P IV having a discrepancy of about a thousand cm−1 , which is
3
P0e
175499.4
175535.4
-36
3
P1e
175572.8
175608.1
-35
3
P2e
175699.0
175732.9
-34
1
D2e
188899.9
188882.5
17
1
S0e
235421.9
235369.3
53
TABLE IV: Relative contributions of third-order two-body
terms for C III.
Term (%) Term (%) Term (%) Term
Z
0.1 D1
-0.2 D6
0.0 T1
S1
0.1 D2
0.4 D7
0.0 T2
S2
-0.4 D3
-1.6 D8
0.3 T3
S3
-0.1 D4
0.1 D9
-0.1 T4
S4
0.4 D5
0.2 D10 -1.0 Q
(%)
4.2
-0.3
-0.5
2.0
41.2
Term (%)
B1
-0.8
B2
-0.1
B3
-46.1
TABLE V: Relative contributions of third-order two-body
terms for P IV.
Term (%) Term
Z
0.1 D1
S1
0.4 D2
S2
-0.3 D3
S3
0.2 D4
S4
1.4 D5
(%)
-0.4
0.3
-3.6
0.6
-1.1
Term
D6
D7
D8
D9
D10
(%)
-0.1
-0.3
0.1
-0.5
-5.0
Term (%) Term (%)
T1
9.4 B1
-3.9
T2
0.1 B2
-0.7
T3
2.1 B3
-29.2
T4
10.2
Q
29.9
somewhat larger, but comparable, to the accuracy of our
third-order calculations.
It is informative to analyze our results in terms of diagrams. The relative contributions of the third-order
two-body terms for the ground-state energy of a typical member in the Be sequence and the ion P IV are
summarized in Tables IV and V. Dominant classes of
diagrams are Q and B3 . This is understandable since
Q are quadruple excited-state diagrams and involve no
core excitations. The quadruple excitation diagrams are
entirely due to valence-valence correlation effects; they
are expected to be large because of the strong repulsion
of the outer valence electrons. Class B3 are backwards
diagrams, which are characteristic of open-shell systems.
As shown in Fig. 1, this class is also associated solely
with valence-valence correlation. The two classes Q and
B3 tend to cancel each other as there is an extra phase of
-1 associated with backwards diagrams. It is interesting
to note that even after subtraction of the contributions
from Q and B3 , their difference is still larger than the
contribution from any other single class of diagrams for
C III.
6
IV.
CONCLUSIONS
The accuracy of third-order MBPT results is at 0.2%
level for lightly-charged ions of both Be and Mg isoelectronic sequences. This level of accuracy is comparable
or superior to the two ab initio methods mentioned in
Sec. III. A complete third-order calculation is important
to understand the relative importance of different contributions to energies of divalent systems. The folded
diagrams as well as the quadruple excitation diagrams
are significant. The dominant role of these two classes
of diagrams is attributed to the strong correlation of the
two valence electrons. This conclusion is useful for workers developing combined CI-MBPT methods which include dominant third-order diagrams. It is also helpful
for researchers setting up SDCC calculations as they try
to classify and account for the contributions from the
third-order diagrams associated with omitted triple excitations. Although one might expect a complete fourthorder calculation for divalent systems to improve the accuracy of the present calculations still further, it is unlikely that such a complex calculation will be carried out
in the near future.
APPENDIX
Angular decompositions of direct formulas for the
third-order two-body part are presented.
Z =
X
XL1 (cdwv)XL2 (abcd)ZL3 (w0 v 0 ab)
(εab − εv0 w0 ) (εcd − εv0 w0 )
L1 L2 L3
abcd
S1 =
×(−1)J+L1 +L2 +L3 +ja +jb +jc +jd +jw0 +jv
(
)(
)(
)
J jc jd
ja jb J
J j a jb
×
L1 jv jw
j d j c L2
L3 j v 0 j w 0
1 X XL (acmw)ZL (mbac)ZL0 (v 0 w0 vb)
[jw ]
(εbv − εv0 w0 ) (εacv − εmv0 w0 )
LL0
abcm
0
×(−1)J+L
+ja +jb +jc +jm +jw0 +jv
1
×δjb jw
[L]
S2 =
)
(
J jv 0 j w 0
L0 jb jv
X ZL (abwm)ZL (mcbv)ZL0 (v 0 w0 ca)
(εac − εv0 w0 ) (εabv − εmv0 w0 )
0
LL
abcm
0
S3 =
×(−1)1+L +ja +jb +jc +jm +jw0 +jv
(
)
)(
1
J ja jc
J jc ja
×
[L] L jv jw
L0 jw0 jv0
X XL (abwm)ZL (w0 cab) ZL (v 0 mvc)
1
2
abcm
(
×(−1)
)
L1 L2 L3
jc jm jb
)(
)
L3
J j v 0 jw 0
ja
L3 jw jv
J+jv +jw
(
jw jw 0
L2 L1
X ZL (abwm)ZL (v 0 cvb) ZL (w0 mac)
1
2
3
(εac − εmw0 ) (εabv − εmv0 w0 )
×
S4 =
3
(εcv − εmv0 ) (εabv − εmv0 w0 )
L1 L2 L3
L1 L2 L3
abcm
(
×(−1)
J+L1 +L2 +L3 +jv +jw
D1
LL
The work of H.C.H. and W.R.J. was supported in part
by National Science Foundation (NSF) Grant No. PHY04-56828. The work of M.S.S. was supported in part by
NSF Grant No. PHY-04-57078.
)
)(
L1
j w j w 0 L2
×
jc
L3 L1 ja
1 X XL (abmn)ZL (mnwb)ZL0 (v 0 w0 va)
= −
[jw ]
(εbw − εmn ) (εav − εv0 w0 )
0
(
Acknowledgments
J jv0 jw0
L2 j w j v
)
L2 L3
jm jb
abmn
0
×(−1)J+L
1
×δja jw
[L]
+ja +jb +jm +jn +jw0 +jv
)
(
J j v 0 jw 0
L0 ja jv
7
D2 =
X
XL1 (abmn)XL2 (mnvw)ZL3 (v 0 w0 ab)
(εabvw − εmnv0 w0 )
0
L1 L2 L3
abmn
»
×
D3
1
1
+
− εmn )
(εab − εv0 w0 )
(εvw
D9 = −
L1 L2 L3
×(−1)
×
1
1
+
(εab − εnw0 )
(εvw − εmv0 )
×(−1)
D10
×δjm jw0
D4
1
[L]
–
jw 0 jw
L3 L1
1 X 0 XL (w0 anm) ZL (mnar)ZL0 (v 0 rvw)
=
[jw0 ]
(εavw − εmnv0 ) (εvw − εrv0 )
0
×
J j v 0 jm
L0 j w j v
T1
LL
amnr
0
L
×(−1)J+L+ja +jb +jm +jn +jw0 +jv
(
)
1
J jw0 jv0
× 2
[L]
L j v jw
X ZL (w0 amn) ZL (nbaw)ZL0 (v 0 mvb)
= −
(εavw − εmnv0 ) (εbv − εmv0 )
0
T2
×(−1)J+1+L +ja +jm +jn +jv
(
)
1
J jv0 jw0
×δjr jw0
[L] L0 jw jv
X XL (w0 amn) ZL (mnwr)ZL (v 0 rva)
1
2
3
=
(εavw − εmnv0 ) (εav − εrv0 )
L1 L2 L3
amnr
LL
×(−1)J+1+L+ja +jb +jm +jn +jw0 +jv
)
)(
(
1
j w 0 j w L0
J j v 0 jw 0
×
[L] L0 jw jv
jb jm L
X ZL (w0 amn) ZL (nbvw)ZL (v 0 mab)
3
T3 =
ZL (w0 anr) ZL (v 0 rma) ZL0 (mnvw)
(εavw − εnrv0 ) (εvw − εmn )
0
abmn
T4
L1 L2 L3
amnr
(
L1 L2 L3
jn ja jm
)
)(
L3
J j v 0 jw 0
jb
L3 j w j v
×(−1)
)
(
J+L1 +L2 +L3 +jv +jw
(
D8
)
j v 0 jw 0
jw jv
×(−1)1+L +ja +jm +jn +jr +jw0 +jv
(
)(
)
1
J jn jm
J jm jn
×
[L] L jv0 jw0
L0 jw jv
0
X ZL (w arn) ZL (v 0 nvm) ZL (rmwa)
1
2
3
=
(εavw − εnrv0 ) (εaw − εmr )
L1 L2 L3
×
0
LL0
×(−1)1+L1 +L2 +L3 +jv0 +jv
8
9
(
)> J j
n jb >
<
=
J j n jb
×
jw0 L1 jm
>
L2 j w j v >
:
;
j v 0 j a L3
X ZL (abmw)ZL (w0 mbn) ZL (v 0 nva)
1
2
3
= −
(εabv − εmv0 w0 ) (εav − εnv0 )
×(−1)
X
)
L3
jn
amnr
abmn
D7
jw 0 jw
L2 L1
×
L1 L2
jr ja
)(
L3
J
L3
jm
J+jv +jw
(
(εavw − εmnv0 ) (εab − εmv0 )
L1 L2 L3
(
×(−1)
abmn
D6 =
)
L1 L2 L3
ja jn jb
)(
)
L2
J j v 0 jw 0
jm
L2 jw jv
(
)
2
(
×(−1)J+jv +jw
X ZL (abmn)ZL (w0 nwb) ZL (v 0 mva)
=
(εbw − εnw0 ) (εav − εmv0 )
1
)
L3
jm
)
jw jw 0
j v 0 jw 0
×
L3 L1
jw jv
X XL (w0 bmn) ZL (v 0 avb) ZL (mnwa)
1
2
3
= −
(εbvw − εmnv0 ) (εaw − εmn )
abmn
abmn
D5
L1 L2
jn jb
)(
L2
J
ja
L2
L1 L2 L3
J+L0 +ja +jb +jm +jn +jw0 +jv
(
(
J+jv +jw
(
LL
»
XL1 (abwm)ZL2 (v 0 mvn) ZL3 (w0 nab)
(εabv − εmv0 w0 ) (εab − εnw0 )
abmn
–
×(−1)J+L1 +L2 +L3 +ja +jb +jm +jn +jw0 +jv
(
)(
)(
)
ja jb J
J j m jn
J ja jb
×
jn jm L1
L2 j w j v
L3 jw0 jv0
0
X
0 ZL (abmn)XL (w nab) ZL0 (v 0 mvw)
1
= −
[jw0 ]
(εabvw − εmnv0 w0 )
0
abmn
X
jw jw0
L2 L1
X ZL (w0 bwn) ZL (nabm)ZL (v 0 mva)
=
(εbv − εnv0 ) (εav − εmv0 )
L
abmn
×(−1)J+L+ja +jb +jm +jn +jw0 +jv
)
(
1
J jw0 jv0
× 2
[L]
L j v jw
)
L1 L2 L3
jm ja jn
)(
)
L2
J j v 0 jw 0
jr
L2 jw jv
jw 0 jw
L3 L1
X 0 XL (v 0 w0 rs) XL (rsmn)ZL (mnvw)
1
2
3
(εvw − εrs ) (εvw − εmn )
×
Q =
(
J+L1 +L2 +L3 +jv +jw
L1 L2 L3
mnrs
×(−1)J+L1 +L2 +L3 +jm +jn +jr +js +jw0 +jv
)
)(
)(
(
J jm j n
jr js J
J j r js
×
L3 jw jv
j n j m L2
L1 j w 0 j v 0
8
B1 = −
X ZL (w0 amn) XL (nmax)ZL0 (v 0 xvw)
[jw0 ]
(εavw − εmnv0 ) (εax − εmn )
0
1
LL
amnx
×(−1)
J+1+L0 +ja +jm +jn +jv
×δjx jw0
B2 = −
where [ji ] ≡ 2ji + 1 is the occupation number of shell i,
and
(
1
[L]
)
J jv0 jw0
L0 jw jv
X ZL (v 0 axm) ZL (w0 mya) ZL0 (xyvw)
(εavw − εmxw0 ) (εay − εmw0 )
0
LL
amxy
(
e
Π (li , lk , L) =
1 if li + lk + L is even
.
0 if li + lk + L is odd
The Slater integral RL (ijkl) is
1+L0 +ja +jm +jx +jy +jw0 +jv
B3
×(−1)
)(
(
)
1
J j x jy
J jx jy
×
[L] L jw0 jv0
L0 j w j v
X 0 XL (v 0 w0 mn) XL (mnxy)ZL (xyvw)
1
2
3
= −
(εvw − εmn ) (εxy − εmn )
L1 L2 L3
mnxy
×(−1)J+L1 +L2 +L3 +jm +jn +jx +jy +jw0 +jv
(
)(
)(
)
J jm jn
jm jn J
J j x jy
×
.
L1 j w 0 j v 0
j y j x L2
L3 jw jv
Z
∞Z ∞
RL (ijkl) =
dr1 dr2
0
0
[Pi (r1 )Pk (r1 ) + Qi (r1 )Qk (r1 )]
× [Pj (r2 )Pl (r2 ) + Qj (r2 )Ql (r2 )] .
The quantity ZL (ijkl) is defined by
The effective interaction strength
­ ° ° ®­ ° ° ®
XL (ijkl) = (−1)L i °C L ° k j °C L ° l RL (ijkl),
ZL (ijkl) ≡ XL (ijkl) + [L]
is independent of magnetic quantum numbers. The reL
duced matrix element of the C tensor is
Ã
!
p
­ ° L° ®
ji jk L
ji + 21
°
°
[ji ] [jk ]
i C k = (−1)
Πe (li , lk , L) ,
− 12 12 0
[1] A. Ivanova, U. Safronova, and V. Tolmachev, Litov.
Phys. Sb. 7, 571 (1967).
[2] U. Safronova and A. Ivanova, Opt. Spectrosc 27, 193
(1969).
[3] E. Ivanova and U. Safronova, J. Phys. B 8, 1591 (1975).
[4] T. N. Chang, Phys. Rev. A 39, 4946 (1989).
[5] T. N. Chang, Phys. Rev. A 34, 4550 (1986).
[6] C. Fischer, M. Godefroid, and J. Olsen, J. Phys. B 30,
1163 (1997).
[7] F. Galvez, E. Buendia, and A. Sarsa, J. Chem. Phys.
118, 6858 (2003).
[8] K. Cheng, Y. Kim, and J. Desclaux, At. Data Nucl. Data
Tables 24, 111 (1979).
[9] A. Ynnerman and C. Froese-Fisher, Phys. Rev. A 51,
2020 (1995).
[10] P. Jonsson and C. Fischer, J. Phys. B 30, 5861 (1997).
[11] E. Lindroth and J. Hvarfner, Phys. Rev. A 45, 2771
(1992).
[12] M. Chen, K. Cheng, and W. Johnson, Phys. Rev. A 64,
042507 (2001).
[13] M. Chen and K. Cheng, Phys. Rev. A 55, 3440 (1997).
L
r<
L+1
r>
X
L0
(
)
jj jl L
ji jk L0
XL0 (ijlk).
[14] I. Savukov and W. Johnson, Phys. Rev. A 65, 042503
(2002).
[15] M. Safronova, W. Johnson, and U. Safronova, Phys. Rev.
A 53, 4036 (1996).
[16] S. Blundell, W. Johnson, and J. Sapirstein, Phys. Rev.
A 42, 3751 (1990).
[17] J. Sucher, Phys. Rev. A 22, 348 (1980).
[18] M. Mittleman, Phys. Rev. A 4, 893 (1971).
[19] M. Mittleman, Phys. Rev. A 5, 2395 (1972).
[20] M. Mittleman, Phys. Rev. A 24, 1167 (1981).
[21] I. Lindgren and J. Morrison, Atomic Many-Body Theory
(Springer-Verlag, Berlin, 1986), 2nd ed.
[22] Tech.
Rep.,
National
Institute
of
Standards
and
Technology
(2006),
uRL:
http://physics.nist.gov/PhysRefData/ASD/index.html.
[23] K. Cheng, W. Johnson, and J. Sapirstein, Phys. Rev. A
47, 1817 (1993).
[24] J. Sapirstein, Phys. Scr. 46, 52 (1993).
[25] R. Chaudhuri, B. Das, and K. Freed, J. Chem. Phys.
108, 2556 (1998).
9
Z
S1
S2
S3
S4
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
T1
T2
T3
T4
Q
B1
B2
B3
FIG. 1: Third-order Goldstone diagrams (two-body part).